Matrix Elements Research Articles

Decoherence is an Echo of Anderson Localization in Open Quantum Systems

Abstract We study the time evolution of single-particle quantum states described by a Lindblad master equation with local terms. By means of a geometric resolvent equation derived for Lindblad generators, we establish a finite-volume-type criterion for the decay of the off-diagonal matrix elements in the position basis of the time-evolved or steady states. This criterion is shown to yield exponential decay for systems where the non-hermitian evolution is either gapped or strongly disordered. The gap exists, for example, whenever any level of local dephasing is present in the system. The result in the disordered case can be viewed as an extension of Anderson localization to open quantum systems.

Multiscale study on the compressive performance of diverse solid waste backfill bodies

To create green, low-energy self-compacting backfill materials, this study utilizes aeolian sand, slag, red mud, and calcium carbide slag to prepare diverse solid waste backfill materials in synergy. Additionally, a suitable amount of polypropylene fibers is added to enhance its toughness. During the experimental process, unconfined compressive strength tests were conducted using digital speckle technique. Computerized tomography scanning technology was employed for internal 3D visualization of the samples. Material strength difference mechanisms were characterized using SEM–EDS and XRD microscopic techniques. Finally, discrete element numerical simulation was utilized for analyzing the evolution of sample failure. The results show that with the increase in calcium carbide slag doping, the specimen UCS increases and then decreases, and when the doping of calcium carbide slag is 14%, the specimen UCS reaches a local maximum of 3.552 MPa; with the increase in the water–solid ratio, the specimen UCS gradually decreases, and when the water–solid ratio is 0.3, the specimen UCS reaches a local minimum of 3.26 MPa. With the increase in calcium slag doping, the mass percentage of calcium element and calcium-silicon ratio in the multi-solid waste matrix first increased and then decreased, and the micro-morphology of the multi-solid waste matrix included lamellar, block, and rod structures; With the increase in the water–solid ratio, the micro-morphology of the multi-solid waste matrix was gradually transformed, and the pore defects between the cementitious materials gradually increased. With the increase in calcium carbide slag doping, the porosity within the unit decreases, and when the doping of calcium carbide slag exceeds 14%, the porosity increases and the compressive strength decreases; with the gradual increase in the water–solid ratio, the porosity within the unit gradually increases, and the number of holes in the specimen reaches a minimum of 208 within the range of the study when the water–solid ratio is 0.24. Digital speckle technique and PFC numerical simulation together elucidate the sample’s failure process and crack evolution mechanism. Initially, the internal voids of the sample are gradually compacted, and crack propagation is slow. As the load increases to the peak, localized deformation and macroscopic crack formation occur in the sample. After the peak load, the sample exhibits significant displacement deformation, and the number of cracks increases significantly. The research findings can provide reference for the design and construction of solid waste backfill pipelines and trenches.

Distance and Orientation Dependence of Triplet-Triplet Energy Transfer Couplings Based on Nonorthogonal Multireference Wave Functions.

Triplet-triplet energy transfer (TEnT) is of particular interest in various photochemical, photobiological, and energy science processes. It involves the exchange of spin and energy of electrons between two molecular fragments. Here, quasi-diabatic self-consistent field solutions were used to obtain the diabatic states involved in TEnT. The resonant Hartree-Fock approach was used to compute the nonorthogonal matrix elements for the two-state or four-state effective Hamiltonian and the overlap matrix. From the symmetric orthogonalized Hamiltonian, electronic coupling elements between the diabatic states in the TEnT process can be obtained. Two structural models, namely, naphthalene dimer and the 2,2'-bifluorene molecule, were employed to investigate the role of distance and orientation of the molecular fragments on the energy transfer process. It is observed that the inclusion of charge transfer states is critical to obtain the correct description of TEnT couplings. We discuss the effectiveness of the two-state model and four-state model in the successful evaluation of TEnT couplings. Spin density plots and biorthogonal orbitals were utilized to verify that the correct diabatic electronic structure of the TEnT states was determined.

A proof of $${\mathfrak {L}}^2$$-boundedness for magnetic pseudodifferential super operators via matrix representations with respect to parseval frames

Abstract A fundamental result in pseudodifferential theory is the Calderón–Vaillancourt theorem, which states that a pseudodifferential operator defined from a Hörmander symbol of order 0 defines a bounded operator on $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) . In this work we prove an analog for pseudodifferential super operator, i. e. operators acting on other operators, in the presence of magnetic fields. More precisely, we show that magnetic pseudodifferential super operators of order 0 define bounded operators on the space of Hilbert–Schmidt operators $${\mathfrak {L}}^2 \bigl ( {\mathcal {B}} \bigl ( L^2({\mathbb {R}}^d) \bigr ) \bigr )$$ L 2 ( B ( L 2 ( R d ) ) ) . Our proof is inspired by the recent work of Cornean et al. (J Fourier Anal Appl 30(21):1–21, 2024. https://doi.org/10.1007/s00041-024-10072-4) and rests on a characterization of magnetic pseudodifferential super operators in terms of their “matrix elements” computed with respect to a Parseval frame.

Observation of a rare beta decay of the charmed baryon with a Graph Neural Network

The beta decay of the lightest charmed baryon Λc+ provides unique insights into the fundamental mechanism of strong and electro-weak interactions, serving as a testbed for investigating non-perturbative quantum chromodynamics and constraining the Cabibbo-Kobayashi-Maskawa (CKM) matrix parameters. This article presents the first observation of the Cabibbo-suppressed decay Λc+→ne+νe, utilizing 4.5 fb−1 of electron-positron annihilation data collected with the BESIII detector. A novel Graph Neural Network based technique effectively separates signals from dominant backgrounds, notably Λc+→Λe+νe, achieving a statistical significance exceeding 10σ. The absolute branching fraction is measured to be (3.57 ± 0.34stat. ± 0.14syst.) × 10−3. For the first time, the CKM matrix element Vcd is extracted via a charmed baryon decay as 0.208±0.011exp.±0.007LQCD±0.001τΛc+. This work highlights a new approach to further understand fundamental interactions in the charmed baryon sector, and showcases the power of modern machine learning techniques in experimental high-energy physics.

The Properties of the Low-Lying Electronic States and Avoided Crossings of the SbP Molecule: A Theoretical Investigation Includes Spin-Orbit Coupling.

High-level multireference configuration interaction plus Davidson correction (MRCI + Q) calculation method was employed to determine the potential energy curves (PECs) of 10 Λ-S states, which come from the first and second dissociation channels of the SbP molecule, as well as 34 Ω states considering the spin-orbit coupling (SOC) effect. By solving the Schrödinger equation for nuclear motion, spectroscopic constants for the ground state X1Σ+ and low-lying excited states were obtained and compared with experimental data. The excellent agreement indicates the reliability of our calculations. Additionally, the calculated spin-orbit (SO) matrix elements of the 13Π and 15Π states with other Λ-S states were analyzed, and the majority of the values in the Franck-Condon region exceed 200 cm-1, indicating strong interactions between these states. What's more, the joint effects of spin-orbit coupling and avoided crossing were discussed in detail, leading to the complex potential energy curves and double-well phenomena observed in the Ω states. Taking forbidden transitions into account, transition dipole moments with the SOC effect are considered. The Franck-Condon factors, Einstein coefficients, and radiative lifetimes for the 13Σ+1 ↔ X1Σ+0+ transition were obtained. Analysis indicates that direct laser cooling of SbP is inappropriate.

Measurement of tt¯ production in association with additional b-jets in the eμ final state in proton–proton collisions at s = 13 TeV with the ATLAS detector

This paper presents measurements of top-antitop quark pair (tt¯) production in association with additional b-jets. The analysis utilises 140 fb−1 of proton–proton collision data collected with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of 13 TeV. Fiducial cross-sections are extracted in a final state featuring one electron and one muon, with at least three or four b-jets. Results are presented at the particle level for both integrated cross-sections and normalised differential cross-sections, as functions of global event properties, jet kinematics, and b-jet pair properties. Observable quantities characterising b-jets originating from the top quark decay and additional b-jets are also measured at the particle level, after correcting for detector effects. The measured integrated fiducial cross-sections are consistent with tt¯bb¯ predictions from various next-to-leading-order matrix element calculations matched to a parton shower within the uncertainties of the predictions. State-of-the-art theoretical predictions are compared with the differential measurements; none of them simultaneously describes all observables. Differences between any two predictions are smaller than the measurement uncertainties for most observables.

Semileptonic decay of the triply heavy Ωccb to the observed Ξcc++ state

We investigate the weak semileptonic decay of the Ωccb+→Ξcc++ℓν¯ℓ, where a triply heavy baryon with spin 1/2 decays into the observed doubly heavy baryon with spin 1/2, using quantum chromodynamics (QCD) sum rule method in all lepton channels. We compute the six relevant vector and axial vector form factors entering the low energy matrix elements in full theory. The invariant form factors are building blocks, using the fit faction of which in terms of q2 in whole physical region, we calculate the exclusive widths in three lepton channels. Our predictions may help the present and future experiments in the course of their search for doubly and triply heavy baryons. Published by the American Physical Society 2025

Circularly polarized RABBITT on atomic shells with large orbital momentum

Abstract In light atoms, the technique of Reconstruction of Attosecond Bursts by Interference of Two-Photon Transitions (RABBITT), driven by circularly polarized radiation, enables a complete retrieval of ionization amplitudes in the XUV+IR regime, including their relative phases [Phys. Rev. Res. 6, L012002 (2024)]. In this work, we extend this method to heavier atoms by conducting numerical simulations of circularly polarized RABBITT on the Kr 3d and Xe 4d atomic shells. The XUV ionization dynamics in these atoms differ markedly due to the presence of a giant shape resonance in the 4d → Ef ionization channel of Xe, a feature absent in Kr. Nevertheless, when an IR photon is added to the XUV ionization, the results for Kr and Xe are surprisingly similar, with the ratio of absorption/emission matrix elements and their relative phases remaining close in value. Moreover, these ratios and phases show minimal variation even when the Xe atom is encapsulated within a C60 fullerene cage. In search for the Cooper-like minimum in the IR absorption amplitude ratios predicted in [J. Phys. B 57, 235601 (2024)], we extend our investigation to the Cr 4f shell where this minimum is indeed present.

An a priori method for estimating the informativeness of the configuration of sensor placement when solving inverse problems of remote sensing

Abstract The question of finding the optimal location and number of sensors is important when solving applied inverse problems of remote sensing. An incorrect answer to this question can significantly affect (1) the accuracy of reconstructing the unknown parameters of the object under study, and/or (2) the computational complexity of the inverse problem being solved. The work proposes an economical algorithm that allows among all equivalent (in the sense of the computational complexity of obtaining a solution to the inverse problem) sensor configurations, to select the one that will give a more accurate result when processing experimental data. This algorithm also allows to make a conclusion about whether it is really necessary to process all the experimental data obtained in the experiment, or whether it is possible to limit ourselves to only part of them without a significant loss of accuracy in the reconstructed solution. The algorithm is based on (1) the application of the mosaic-skeleton approximation method to the matrix of the system of equations to which the inverse problem being solved is reduced, and (2) the calculation of the compression rate of the approximating matrix relative to the original dense matrix for a given permissible relative approximation error. Thus, the algorithm is a priori, that is, it does not require a preliminary search for a solution to the inverse problem for the sensor configuration under study. Moreover, if the formulas that define the elements of the system matrix are known, then the algorithm requires the calculation of not all of these elements, but only part of them.

HELP: Everlasting Privacy through Server-Aided Randomness

Everlasting (EL) privacy offers an attractive solution to the Store-Now-Decrypt-Later (SNDL) problem, where future increases in the attacker's capability could break systems which are believed to be secure today. Instead of requiring full information-theoretic security, everlasting privacy allows computationally-secure transmissions of ephemeral secrets, which are only "effective" for a limited periods of time, after which their compromise is provably useless for the SNDL attacker. In this work we revisit such everlasting privacy model of Dodis and Yeo (ITC'21), which we call Hypervisor EverLasting Privacy (HELP). HELP is a novel architecture for generating shared randomness using a network of semi-trusted servers (or "hypervisors"), trading the need to store/distribute large shared secrets with the assumptions that it is hard to: (a) simultaneously compromise too many publicly accessible ad-hoc servers; and (b) break a computationally-secure encryption scheme very quickly. While Dodis and Yeo presented good HELP solutions in the asymptotic sense, their solutions were concretely expensive and used heavy tools (like large finite fields or gigantic Toeplitz matrices). We abstract and generalize the HELP architecture to allow for more efficient instantiations, and construct several concretely efficient HELP solutions. Our solutions use elementary cryptographic operations, such as hashing and message authentication. We also prove a very strong composition theorem showing that our EL architecture can use any message transmission method which is computationally-secure in the Universal Composability (UC) framework. This is the first positive composition result for everlasting privacy, which was otherwise known to suffer from many "non-composition" results (Müller-Quade and Unruh; J of Cryptology'10).

False Strange Attractors as Sources of Pseudo Randomness

Abstract In this work, the use of false chaotic attractors in enhancing the security of chaos-based encryption applications will be explored. False attractors result by rearranging the solution trajectories of a nonlinear dynamical system, leading to novel shapes in the 3D space. This way, a single solution trajectory of a chaotic system can be used to generate numerous new false trajectories. Examples of such attractors are provided for the Lorenz system, and a system with line equilibrium from Moysis et al. (2019). By formalizing the method of constructing false attractors, and generalizing it by adding a series of elementary operations, the key parameters of this procedure can be used as part of the secret key in symmetric encryption applications, effectively increasing the key space. Specifically, the key space increases significantly with respect to the simulation time, and the system’s dimension. Moreover, due to its fast implementation, this technique can be used as a less complex, but much lighter alternative to full scale permutation. The usability of the false attractors as sources of randomness is showcased through the design of a chaotic pseudo-random bit generator. The generator can output 64 bits per iteration, and is equally effective under any false attractor of the chaotic system.

Dynamic Phosphorescence Behavior of Carbene-Metal-Amide Complexes from the Perspective of Excited State Modulation.

Carbene-metal-amide (CMA) complexes have diverse applications in luminescence, imaging and sensing. In this study, we designed and synthesized a series of CMA complexes, which were subsequently doped into a PMMA host. These materials demonstrate light-induced dynamic phosphorescence, attributed to their long intrinsic triplet state lifetime (τP,int, in the μs-ms scale), high intersystem crossing (ISC) rate constant (kISC, up to 107 s-1), and bright phosphorescence. The extended τP,int, and elevated kISC facilitate efficient sensitization of singlet oxygen (1O2) under light irradiation, which is rapidly consumed by the host material, creating a localized anaerobic environment conducive to bright phosphorescence emission. The Sn-T1 process exhibits a large spin-orbital coupling matrix element (SOCME) value, while the SOCME value between T1 and S0 is comparatively smaller, resulting in a large kISC and long τP,int, Computational results indicate that the hole-electron configuration in the lowest triplet state exhibits low contributions from gold. Based on the dynamic phosphorescence properties, an encryption material capable of achieving a "burn after reading" effect was developed. This work illustrates that those phosphorescent emitters with minimal heavy atom contribution can produce dynamic phosphorescent phenomena, providing a novel strategy for designing stimuli-responsive phosphorescent materials.

Higher partial wave contamination in finite-volume 1-to-2 transitions

In their seminal work, Lellouch and Lüscher derived a conversion factor relating a finite-volume matrix element, calculable using numerical lattice QCD, with the infinite-volume decay amplitude for K → ππ. The conversion factor depends on the ππ → ππ scattering amplitude with the same total isospin as the decay channel (either zero or two). Although an infinite tower of ππ → ππ partial-wave components affect the conversion factor, the S-wave (ℓ = 0) component is expected to dominate, and only this contribution is included in the well-known Lellouch-Lüscher factor, with other ππ → ππ partial-wave amplitudes formally set to zero. However, as the precision of lattice calculations increases, it may become important to assess the systematic uncertainty arising from this approximation. With this motivation, we compare the S-wave-only results with those truncated at the next contaminating partial wave: the G-wave (ℓ = 4) for zero total momentum in the finite-volume frame and the D-wave (ℓ = 2) otherwise. Using the general framework for 1→J2 transitions derived in ref. [1], we quantify the effect of higher partial waves for systems with zero and non-zero total momentum as well as with anti-periodic boundary conditions, presenting both generic numerical examples and results for realistic ππ amplitudes taken from chiral perturbation theory and dispersive analysis. We also consider the accidental degeneracy occurring in the 8th excited state of the zero-momentum system. This exhibits qualitatively new features at ℓ = 4, not seen in the ℓ = 0 truncation.

Impact of Dipole Self-Energy on Cavity-Induced Nonadiabatic Dynamics.

The coupling of matter to the quantized electromagnetic field of a plasmonic or optical cavity can be harnessed to modify and control chemical and physical properties of molecules. In optical cavities, a term known as the dipole self-energy (DSE) appears in the Hamiltonian to ensure gauge invariance. The aim of this work is twofold. First, we introduce a method, which has its own merits and complements existing methods, to compute the DSE. Second, we study the impact of the DSE on cavity-induced nonadiabatic dynamics in a realistic system. For that purpose, various matrix elements of the DSE are computed as functions of the nuclear coordinates and the dynamics of the system after laser excitation is investigated. The cavity is known to induce conical intersections between polaritons, which gives rise to substantial nonadiabatic effects. The DSE is shown to slightly affect these light-induced conical intersections and, in particular, break their symmetry.

Distance and Voltage Dependence of Orbital Density Imaging Using a CO-Functionalized Tip in Scanning Tunneling Microscopy.

The appearance of frontier molecular ion resonances measured with scanning tunneling microscopy (STM)─often referred to as orbital density images─of single molecules was investigated using a CO-functionalized tip in dependence on bias voltage and tip-sample distance. As model systems, we studied pentacene and naphthalocyanine on bilayer NaCl on Cu(111). Absolute tip-sample distances were determined by means of atomic force microscopy (AFM). STM imaging revealed a transition from predominant p- to s-wave tip contrast upon increasing the tip-sample distance, but the contrast showed only small changes as a function of voltage. The distance-dependent contrast change is explained with the steeper decay of the tunneling matrix element for tunneling between two p-wave centers, compared to tunneling between two s-wave centers. In simulations with a fixed ratio of s- to p-wave tip states, we can reproduce the experimental data including the distance-dependent transition from predominant p- to s-wave tunneling contribution.

Elementary-level intrusive coupling of machine learning for efficient mechanical analysis of variable stiffness composite laminates: a spatially-adaptive fidelity-sensitive computational framework

AbstractMechanical analysis of the complex configurations of composite laminates can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations since the error accumulates with multiple realizations, leading to poor outcomes. We propose an elementary-level coupling of machine learning for efficient, yet accurate mechanical analysis of laminated composites based on finite element simulations coupled with gaussian process regression. The generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates are accounted for forming an efficient mapping with the augmentation of lower-order theory-based elementary-level structural matrices. The computationally efficient machine learning models predict the difference in the elements of the stiffness matrix for higher-order zigzag theory (HOZT) and first-order shear deformation theory (FSDT) at the first stage. Based on such machine learning-based difference mapping, we augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory without any additional computational expenses (referred to here as augmented FSDT, or aFSDT). However, it is not necessary to augment all the elements in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we further propose spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. The selective augmentation strategy essentially brings in a scope of integrating physics-based insights of critical stress resultant distribution into the algorithm based on best theory diagram. Subsequently, the global structural matrices are computed exploiting such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency by augmenting selective elements while minimizing the loss of accuracy due to the involvement of surrogates. Detailed numerical results are presented for static, dynamic and stability characterization of composite laminates including the demonstration for variable stiffness composite configurations based on the efficient machine learning-assisted elementary-level intrusive computational framework, wherein the notion of engineering judgement is introduced concerning the trade-off between computational efficiency and required level of accuracy.

Measurement Analysis of Coherence in Femtosecond Laser-Induced Molecular Alignment

Femtosecond laser-induced excitation of molecular rotational states leads to phenomena such as alignment and orientation, which fundamentally stem from the coherence between the induced rotational states. In recent years, the quantitative study of coherence in the field of quantum information has garnered widespread attention. Different kinds of coherence measures have been proposed and investigated. This paper delves into the quantitative correlation between the intrinsic coherence measurement and the degree of molecular alignment induced by femtosecond laser pulses at finite temperatures. By examining the molecular alignment induced by ultrafast non-resonant laser pulses, the study establishes a quantitative relationship between the \(l_1\) norm coherence measure (\(C_{l_1}(\rho)\) -- the sum of the absolute values of all off-diagonal elements of the density matrix \(\rho\)) and the alignment amplitude ($\mathcal{D}\langle \cos^2 \theta \rangle$--the difference between the maximum and minimum alignment). A quadratic relationship $ C_{l_1} = (a + b\mathcal{E}^2_0)\times \mathcal{D}\langle \cos^2 \theta \rangle$ between the the \(l_1\) norm coherence measure and $\mathcal{D}\langle \cos^2 \theta \rangle$ with respect to the electric field intensity $\mathcal{E}_0$ is obtained. This relationship is validated through numerical simulations of the CO molecule and the ratio coefficients $a$ and $b$ for different temperatures are demonstrated. Furthermore, a mapping relationship between this ratio and the pulse intensity area is established. The findings of this study offer an alternative methodology for experimentally detecting the coherence measure within femtosecond laser-excited rotational systems, thereby extending the potential applicability of molecular rotational states to study the coherence measure in the field of quantum resources. This will facilitate the interdisciplinary integration of ultrafast strong-field physics and quantum information.

Fully relativistic distorted-wave method for collision excitation process between electron and atom

The electron-atom (ion) collision excitation process is one of the most common inelastic scattering processes. It is of great significant in the fields of astrophysics and laboratory plasma. The relativistic distorted-wave method is a widely used theoretical tool for studying electron-atom (ion) collisions, with the aim of obtaining scattering parameters, such as impact cross sections and rate coefficients.<br>In recent years, we have developed a set of fully relativistic distorted-wave methods and programs for studying the electron-atom collision excitation processes. This method is based on the multi-configuration Dirac-Hartree-Fock (MCDHF) method, along with the corresponding packages GRASP 92/2K/2018 and RATIP. The present paper introduces the calculations of continuum state wave functions, total and differential cross sections, state multipoles, integral and differential Stokes parameters of the radiation photon following the impact excitation processes of polarized electrons and atoms. The influence of electron correlation effects, Breit interaction, and plasma screening effects on the excitation cross sections is discussed. The present methods and programs offer several advantages:<br>(1) In the calculations of the continuum electron wave functions, the direct and exchange interactions between the bound electron and the continuum electron are included. Then, the anti-symmetrized coupling wave functions, which is composed of the continuum electron and the ion wave functions, are utilized as the wave function of the system. This method has been employed to study the low to high energy electron scattering processes.<br>(2) In this method, the target state wave function is obtained using the MCDHF theory and the corresponding GRASP packages. The MCDHF method has the advantage of being able to consider the electron correlation effects, including valence-valence, core-valence, and core-core correlations, as well as the Breit interaction and quantum electrodynamics (QED) effects effect on the target state wave function. Furthermore, the calculation of the collision excitation matrix elements also incorporates the contribution of the Breit interaction. Consequently, the present method combines the advantages of both the MCDHF method and distorted-wave method, making it suitable for studying the scattering processes of highly charged ions. In addition, it facilitates the study of the influence of higher-order effects on the collision dynamics, thereby enabling the obtaining high-precision theoretical data.<br>(3) The current method and program can also be utilized to study the scattering cross section of electron-atom collision excitation processes, as well as the impact of plasma screening effects on collision excitation. Furthermore, the state multipoles, differential Stokes parameters, integral Stokes parameters and orientation parameters of electron-complex atom collision excitation can be studied in detail using the present method and program.

Neutrinoless ββ decay nuclear matrix elements complete up to N2LO in heavy nuclei

Neutrinoless ββ decay nuclear matrix elements complete up to N2LO in heavy nuclei